Problem: What is the extraneous solution to these equations? $\dfrac{x^2 - 35}{x - 10} = \dfrac{x + 21}{x - 10}$
Solution: Multiply both sides by $x - 10$ $ \dfrac{x^2 - 35}{x - 10} (x - 10) = \dfrac{x + 21}{x - 10} (x - 10)$ $ x^2 - 35 = x + 21$ Subtract $x + 21$ from both sides: $ x^2 - 35 - (x + 21) = x + 21 - (x + 21)$ $ x^2 - 35 - x - 21 = 0$ $ x^2 - 56 - x = 0$ Factor the expression: $ (x + 7)(x - 8) = 0$ Therefore $x = -7$ or $x = 8$ The original expression is defined at $x = -7$ and $x = 8$, so there are no extraneous solutions.